Heat variation on MHD Williamson hybrid nanofluid flow with convective boundary condition and Ohmic heating in a porous material

The aim of the present study is to explore the variation of heat on MHD Williamson hybrid nanofluid (Ag-TiO2/H2O) model for steady two-dimensional and incompressible flow with a convective boundary condition in a curved coordinate porous system with Ohmic heating. Nusselt number is distinguished by the process of thermal radiation. The partial differential equations are controlled by the curved coordinate’s porous system, which depicts the flow paradigm. Employing similarity transformations, the acquired equations were turned into coupled non-linear ordinary differential equations. The governing equations were disbanded by RKF45 via shooting methodology. The focus is on examining physical characteristics such as heat flux at the wall, temperature distribution, velocity of flow, and surface friction coefficient for a variety of related factors. The analysis explained that increasing permeability, Biot and Eckert numbers enhance temperature profile and slowdown heat transfer. Moreover, convective boundary condition and thermal radiation enhance the friction of the surface. The model is prepared as an implementation for solar energy in processes of thermal engineering. Morever, this research has enormous applications in the industries of polymer and glass, also in the field of heat exchangers styling, cooling operations of metallic plates, etc.


Problem formulation
The 2D flux of incompressible and steady Williamson hybrid nanofluid through a stretchable curved linearly plate is studied. The surface is assumed to be wrapped in the shape of a circle with radius R * in the directions of r− and s− , respectively. The velocity is defined as U w (s) = as with extendable constant a > 0 . The process of heat relocation included thermal radiations and convection. The expression of stress-tensor 36 for Williamson fluid is where after employing the approximation of boundary layer, the corresponding formulas for mass conservation, momentum, and energy are 37 : the boundary conditions are: where ν * hnf = µ 0 −µ ∞ ρ hnf . The following similarity transformations are: By using Eq. (9) in Eqs. (4)- (8), we obtain replacing pressure P from Eqs. (10) and (11), we get ∂r + R * ∂(u s ) ∂s = 0, refer to the curvature parameter, Williamson parameter, Reynold number, thermal radiation, parameter of magnetic, porous media permeability, sink/source of heat and Eckert number.
Coefficient of skin friction (c f ) and local Nusselt number (Nu s ) are defined as: where the wall shear stress (τ w ) , and heat flux q w are: eventually, Eq. (15) in nondimensional form become:

Computational procedure
The controlling regime of Eqs. (12) and (13) is connected and highly non-linear. The Runge Kutta Fehlberg (RKF45) cum technique of shooting is applied to solve the system numerically for a variety of parameter values. The action of involved diverse variables on the physical quantities as f ′ (η) , θ(η) , Re 1/2 C f , and Re −1/2 Nu are displayed pictorially. The precision is up to the 5 th decimal point as the criterion of convergence and the step size is taken as �η = 0.01 . Against the boundary condition of far-field, we assumed an acceptable finite value in (14), that is η → ∞ , let us say η ∞ .

Results and discussion
The objective of this part is to go through the impact of diverse factors on flow conduct. Table 1 explains the formulation of hybrid nano-fluid properties that was used. The properties of thermophysical of H 2 O and the nano-particles of Ag/TiO 2 are shown in Table 2. Table 3 depicts the relevance between the previous results and the current results. This indicates the legitimacy of the current results as well the reliability of the numerical approach used in this research. To calculate the approximate relative error ξ between the current findings (r c ) and previous results (r p ) using the formula: ξ = |rc−rp| r c × 100% . Table 4 displays the diverse values of Re 1/2 C f and Re −1/2 Nu for diverse values of ϕ 1 , ϕ 2 , K , Ec , Bi and f w when M = 2.0 , Pr = 6.2 , α 1 = 1.7 , α 2 = 0.1 , R = 0.5 , and S = −0.1 . It has been discovered that ϕ 1 , ϕ 2 and Ec have the same influence on skin friction and Nusselt number. The surface friction is lowered by an improvement of porosity and Biot number, but they increase the number of Nusselt.    Figures 2 and 3 uncover the action of profiles of velocity f ′ (η) and temperature θ(η) being transferred by dimensionless K parameter in Williamson hybrid nano-fluids, respectively. It is seen that f ′ (η) has diminishing conduct for increasing K as illustrated in Fig. 2. The fact behind is that the presence of K leads to upsurge the protection against the fluid's smooth motion which makes f ′ (η) decreases and because of which there is ascend in the distribution of temperature. This conduct of θ(η) is clearly observed from Fig. 3, which explains that the profile of temperature carries out an improvement with the expanding K parameter. Figure 4 exposes the effect of magnetic field on f ′ (η) over a curved extending surface. Since the field of magnetic experiences to be double, the component of the velocity caused to be lowered. The fact behind this is that when M parameter is activated, it creates forces of Lorentz, which resist the fluid flow. Moreover, the momentum boundary layer thickness extremely lowers with an increment in M . So, the parameter of magnetic field has an important turn in the velocity profile. Figure 5 reflects the contrast of θ (η) profile with the field of magnetic. It is detected that increasing happens in temperature with growing values of M . As Lorentz force effects on f ′ (η) causes friction on the flux, this is the main reason for the production of great heat energy.
The temperature distribution under the action of S is portrayed in Fig. 6, when other parameters are fixed. It is undeniable that when S increases, θ(η) improves consistently. Physically, increasing S parameter causes the flux  Bi k hnf k f θ ′ = θ . This implies that θ arrives to 1 as Bi → ∞. Figures 9 and 11 illustrate the influence of nanomolecules of Ag (ϕ 1 ) and TiO 2 (ϕ 2 ) on f ′ (η) respectively, when other parameters are stationary. It is illustrated that growing amounts of ϕ 1 increase f ′ (η) , but increasing values of ϕ 2 reduces the profile of f ′ (η) and the thickness of the corresponding boundary-layer which may be that of greater collision between the suspended nanoparticles. The profiles of temperature under the action of Ag and TiO 2 are portrayed in Figs. 10 and 12. From Figs. 10 and 12 for base liquid and mixture of nanoparticles, any one can observe obviously an improvement in θ (η) with increasing volume fraction of nanoparticle. The   www.nature.com/scientificreports/ truth is that including nanoparticles with various volume fractions improves the thermal characteristics of the steward fluid, hence growing its temperature. For upsurging parameter of R , the profile of temperature lowers initially, whereas reverse demeanor is observed when η > 2.6 . Physically, this is due to an excess in R boosts the increment and transmission of additional heat into the flow, which aids increase the thickness of thermal boundary layer. This conduct of θ(η) is obviously watched from Fig. 13.
The schematics visualizations of the transport of heat conduct due to diverse amounts of S and Ec against M and K are plotted in Figs. 14 and 15, respectively. It is scrutinized that Re −1/2 Nu is improved when S , Ec , M

Conclusions
This study shows the implementation of thermal engineering in the vicinity of solar radiation for the investigation of Williamson hybrid nanofluid over a curved extendable plate with radius R * under the assumption of boundary layer. The curvilinear coordinate model is introduced to model the problem. Significant aspects of change of heat phenomenon have nonlinear radiations, the effects of magnetic field, heat sink/source, Ohmic heating and convective boundary condition are presented in this study. The significant results of this study are:  • Any boost in parameter of heat sink/source causes an increment trend for field of temperature.
• Temperature distribution is upsurged with rising values of M,K,Ec , Bi , R , ϕ 1 and ϕ 2 parameters.
• Increasing coefficient of skin friction has a direct relevance with increasing ϕ 1 , R, Bi.
• Drag force has a reverse function with solid volume fraction of TiO 2 .
• All parameters which we mentioned above have against trend with heat transfer.

Data availability
All data analyzed or generated during this study are included in this article.